Interesting convex function inequality

Question

Let $f:(0,+\infty)\to \mathbb R$ a convex function. Prove that:

$$20\int^3_0 f(x)\,dx + 10\int^6_0 f(x)\,dx \ge 12 \int^5_0 f(x)\,dx + 15 \int^4_0 f(x)\,dx$$ I tried to use the Newton-Leibniz formula but I did not get to any relevant point.

Answer 1

Hint : Apply the inequality $$f(tx+(1-t)y)\leq tf(x)+(1-t)f(y)$$ with $x=3u$, $y=6u$, $t_1=1/3$ and also $t_2=2/3$. Then add, and integrate from $0$ to $1$, change the variables, and to finish multiply by $60$.